A Note on Parametric Kinds of the Degenerate Poly-Bernoulli and Poly-Genocchi Polynomials

: Recently, the parametric kind of some well known polynomials have been presented by many authors. In a sequel of such type of works, in this paper, we introduce the two parametric kinds of degenerate poly-Bernoulli and poly-Genocchi polynomials. Some analytical properties of these parametric polynomials are also derived in a systematic manner. We will be able to ﬁnd some identities of symmetry for those polynomials and numbers.


Introduction
Special functions, polynomials and numbers play a prominent role in the study of many areas of mathematics, physics and engineering. In particular, the Appell polynomials and numbers are frequently used in the development of pure and applied mathematics related to functional equations in differential equations, approximation theories, interpolation problems, summation methods, quadrature rules and their multidimensional extensions (see [1] ).The sequence of Appell polynomials A j (z) can be signified as follows: or equivalently where A(z) = A 0 + A 1 z 1! + A 2 z 2 2! + · · · + A j z j j! + · · · , A 0 = 0, is a formal power series with coefficients A j known as Appell numbers.
The Stirling numbers of the first kind are given by (see, [16][17][18]) and the Stirling numbers of the second kind are defined by (see [19,20]) The degenerate Stirling numbers of the of the second kind are defined by (see [10,21,22]) Note that lim µ−→0 S In the year (2017, 2018), Jamei et al. [23,24] introduced the two parametric kinds of exponential functions as follows (see also [6,[23][24][25]): and where and Recently, Kim et al. [2] introduced the following degenerate type parametric exponential functions: and where and Motivated by the importance and potential applications in certain problems in number theory, combinatorics, classical and numerical analysis and physics, several families of degenerate Bernoulli and Euler polynomials and degenerate versions of special polynomials have been recently studied by many authors, (see [3][4][5][11][12][13]16]). Recently, Kim and Kim [2] have introduced the degenerate Bernoulli and degenerate Euler polynomials of a complex variable. By separating the real and imaginary parts, they introduced the parametric kinds of these degenerate polynomials.
The main object of this article is to present the parametric kinds of degenerate poly-Bernoulli and poly-Genocchi polynomials in terms of the degenerate type parametric exponential functions. We also investigate some fundamental properties of our introduced parametric polynomials.

Parametric Kinds of the Degenerate Poly-Bernoulli Polynomials
In this section, we define the two parametric kinds of degenerate poly-Bernoulli polynomials by means of the two special generating functions involving the degenerate exponential as well as trigonometric functions.
It is well known that (see [2]) The degenerate trigonometric functions are defined by (see [19]) Note that, we have lim In view of Equation (8), we have and From Equations (23) and (24), we note that and p,µ (η, ξ) and degenerate sine-poly-Bernoulli polynomials B (k,s) p,µ (η, ξ) for nonnegative integer p are defined, respectively, by and For η = ξ = 0 in Equations (27) and (28), we get are the new type of poly-Bernoulli polynomials.
Based on Equations (25)-(28), we determine and B (k,s) . (30) and Proof. From Equation (23), we have Similarly, we find In view of Equations (33) and (34), we obtain our first claimed result shown in Equation (31). Similarly, we can establish our second result shown in Equation (32).

Theorem 2.
The following results hold true: and B (k,s) Proof. From Equations (27) and (17), we see Now, by using Equations (27) and (10), we find and B (2,s) Proof. In view of Equation (27), we have Upon setting k = 2, we obtain which gives our required result, Equation (39). The proof of Equation (40) is similar; therefore, we omit the proof.
and B (k,s) Proof. From Equations (27) and (11), we see On using Equation (45) in (44), we find Replacing j by j − r in the right side of above expression and after equating the coefficients of z j , we obtain our needed result, Equation (42). Similarly, we can derive our second result, Equation (43).

Theorem 5.
The following recurrence relation holds true: and B (k,s) Proof. In view of Equation (27), we have which upon replacing j by j − r in the right side of above expression and after equating the coefficients of z j , yields our first claimed result, Equation (46). Similarly, we can establish our second result, Equation (47).
Theorem 6. Let k ∈ Z and j ≥ 0, then we have and B (k,s) Proof. On using Equation (27), we find By comparing the coefficients of z j on both sides, we obtain the result, Equation (48). The proof of Equation (49) is similar to Equation (48).

Theorem 7.
If k ∈ Z and j ≥ 0, then and B (k,s) Proof. From Equations (27) and (12), we find On comparing the coefficients of z j on both sides, we obtain our required result, Equation (50). The proof of Equation (51) is similar to Equation (50).

Parametric Kinds of Degenerate Poly-Genocchi Polynomials
In this section, we introduce the two parametric kinds of degenerate poly-Genocchi polynomials by defining the two special generating functions involving the degenerate exponential as well as trigonometric functions.
In view of Equation (9), we have and From Equations (52) and (53), we can easily get and Definition 2. The degenerate cosine-poly-Genocchi polynomials G (k,c) j,µ (η, ξ) and degenerate sine-poly-Genocchi polynomials G (k,s) j,µ (η, ξ) for nonnegative integer j are defined, respectively, by and On setting η = ξ = 0 in Equations (56) and (57), we get are the new type of poly-Genocchi polynomials. From Equations (54)-(57), we determine Theorem 8. For k ∈ Z and j ≥ 0, we have and Proof. On using Equation (52), we see Similarly, we find By comparing the coefficients of z j on both sides in Equations (62) and (63), we obtain our desired result, Equation (60). The proof of Equation (61) is similar to Equation (60). Theorem 9. If k ∈ Z and j ≥ 0, then and G (k,s) Proof. From Equations (56) and (10), we see r,µ C j−r,µ (η, ξ) Similarly, we find By comparing the coefficients of z j on both sides of Equations (66) and (67)